Efficient prime counting and the Chebyshev primes

Abstract

The function $\epsilon(x)=\mbox{li}(x)-\pi(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $\epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x)$ and $\epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $\mu(n)$ and $\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p<1$, and $j_{\theta(p)}>1$ for $p<10^{11}$. Besides, $j_{\psi(p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set $\mathcal{\mbox{Ch}}$, give accurate approximations of the jump $j_{\psi(p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $\psi(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $\psi(p_n^l)-p_n^l$ (or of the function $\theta(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function $S_N(x)=\sum_{n=1}^N \frac{\mu(n)}{n}\mbox{li}[\psi(x)^{1/n}]$, that is found to be much better than the standard Riemann prime counting function.